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For each entry (prim, j, x, y) in τ , connect the vertices (j − 1, x ⊕ Lj−1 ) and (j, y). For a path P in G(s), let |P | denote the number of edges in this path. ) We deﬁne the following notion of good and bad keys. Definition 2 (Bad and good keys). We say that a key s = (L0 , . . , Lt ) is bad if τ contains an entry (enc, x, y) such that in the graph G(s), there’s a path P0 starting from (0, x) and a path P1 starting from (t, y ⊕ Lt ) such that |P0 | + |P1 | ≥ t. If a key is not bad then we’ll say that it’s good.

Note that if s is good then αk [s] < βk [s] for every k ∈ {1, . . , q}. Recall that in the expectation method, one needs to ﬁnd a non-negative function g : K → [0, ∞) such that g(s) bounds 1 − pS0 (τ, s)/pS1 (τ, s) for all s ∈ Γgood . The function g is directly given in the following technical lemma. The proof, which is based on the main combinatorial lemma of [9], is in Appendix A of the full version of this paper. 3 Note that here the unusual thing is that Case 1 is handled via a direct proof.

We will still resort to the involved and elegant “path-counting” lemma of [9], but it will only be used to deﬁne a function g for which computing the expectation of g(S) will be fairly easy. – Case 2. At least one of q, p1 , . . , pt is bigger than N /4. We’ll use the transcript reduction method, where the other two systems S0 and S1 on which we assume we have established point-wise proximity provide the real and ideal games for a (t − 1)-round KAC. T. Hoang and S. Tessaro Therefore, our proof for Eq.